garethmetcalfe Education General, Maths ideas

Questions for Deepening Part 2

I’ve been on a mission this year to post lots of examples of questions and tasks that I have used on a daily basis to deepen the thinking of the children in my year 6 class. I’m also posting activities that I’ve seen used in classes throughout the school. Below are all my posts from Autumn half-term 2. I hope that there’s something here that you may find useful!

First of all, I found this image from a Chinese textbook showing how > and = signs are introduced:Croc

A task from Y2 that encourages children to use the = sign in different places:
SymbolsA question which is very accessible but can be extended by children working systematically to find all possible solutions:
Addition reasoning

And another allowing children to explore parts and wholes:
KS1

An ‘immersion’ activity used to encourage children to see division as grouping (e.g. representing 26/3 as ‘how many 3s in 26?’):
Division immersion

And some questions that I’ve used with my own class.

Area and perimeter:
area 1 area 2

Percentages:
% 4 % 2 % 1Averages:
Average 2 Average 1 Average 3Below is the visual representation used to unpick that final question:Average 4

My favourite resources for providing deep, meaningful mathematical challenges are those that I have published with Alan Peat ltd. These are more rich, extended and often contextualised tasks.

First Class Maths provides deep, extremely challenging and quirky tasks.

The Maths Apprenticeship gives extended challenges for deepening mathematical and personal competence skills.

Logic Squares is all about getting children playing with numbers.

garethmetcalfe Education General, Maths discussion

Cultural legacy and achievement in maths

This is an exciting time to be involved in maths education, with positive steps being taken to make the curriculum deeper and more conceptual. There’s also a growing awareness that children need to develop a positive self-concept of themselves  as mathematicians and problem-solvers. Our ambition to replicate the mathematical success of East Asian countries has been the driver behind these changes; without doubt there’s a lot that we can learn (and have learnt) from the teaching and learning of maths in these countries.

However, I believe that we can’t look at the success of these Asian nations in mathematics in isolation, without also considering the powerful influence of cultural norms within these countries. This was highlighted very thoughtfully by Malcolm Gladwell in Outliers, a book which studies the roots of success for individuals or groups of people whose achievements sit beyond normal parameters.

Outliers

Gladwell looked specifically at six nations whose results topped the TIMSS international comparison tests for maths – Singapore, South Korea, China, Taiwan, Hong Kong and Japan. Historically, the culture of these five nations were built, Gladwell described, by the tradition and legacy of wet rice agriculture.

Growing rice required extraordinarily precise management of a paddy: irrigation systems needed to be built; water levels had to be precisely managed; the ground must be perfectly flat; seedlings had to be planted and cultivated with great precision. The variability of a yield could be great, and it would be determined by the management of the rice paddy. Gladwell argued that it was this legacy – a culture rooted in the principle of hard, complex work leading to rich rewards – that has led to the phenomenal subsequent success of these Asian countries in mathematics. But how was Gladwell able to draw these conclusions based on the results from the TIMMS tests?

The TIMMS international comparison tests are long and tiresome, being made up of 120 questions. As part of this analysis, the number of questions completed by participants from each country was also analysed. The results were startling. There was an incredibly strong correlation between the number of questions attempted and mathematical attainment. In fact, the results were almost identical: countries with the most successful mathematicians were those who persevered for the longest when completing the test, and vice versa. The data suggested that the willingness to persevere was an unbelievably powerful predictor of success, specifically in maths.

And which countries’ students persevered for the longest? Those whose tradition and culture were shaped by the lessons of rice cultivation. This attitude is neatly exemplified by the Chinese proverb ‘If a man works hard, the land will not be lazy.’ And how richly this principle applies to success in mathematics.

We clearly have a lot to learn pedagogically from the highest performing nations in maths. Importantly, though, I believe that we must also become increasingly proactive in developing positive attitudes towards maths, even if it’s just within our own small circles. After all, we might not be able to control attitudes towards maths at a macro level – but we do have a powerful influence within our own schools. Ultimately, this is a critical factor in determining the extent of children’s achievement in mathematics.

The blog below shows how we have tried to develop a positive mathematical culture within our own school: https://garethmetcalfe.wordpress.com/2014/08/25/establishing-a-mathematical-culture/

garethmetcalfe Education General

Can education learn from memory champions?

Three and a half years ago I was given a book, ‘Moonwalking with Einstein’, as a leaving present from a year 6 child who knew me too well.

IMG_0937.PNG

It chronicles the journey of Joshua Foer, a journalist who became fascinated in the seemingly inhuman achievements of the world’s greatest memory champions. He met serial competitor Ed Cooke, who was convinced that by learning well known memory techniques, and with high quality practice, Josh himself could become a memory champion. Within a year, Josh was transformed from a journalist (with a distinctly average memory) to the U.S. Memory Champion.

The book explains that we can all vastly improve our ability to memorise by converting information into vivid, personal and most importantly visual images. By characterising inanimate objects and information, and visualising these images in familiar settings – memory palaces – we can recall vastly more information.

I trialled these techniques by memorising a 15-item shopping list that my wife wrote during a long motorway journey. I imagined each item being used in various unusual ways during my mother’s 50th birthday party at my childhood family home. More recently, I was able to memorise my class register by picturing the children doing various unusual things in different places within the school.

Without doubt, these memory techniques (essential skills in the ancient world before the widespread use of reading and writing) are freakishly effective for improving our ability to recall information. This leads to the inevitable questions: should these techniques be being taught in schools? And are there other principles from memory training that have an application within education? The techniques are excellent for memorising vast amounts of information, out of context and without understanding. Is there any place for this kind of learning within our schools?

Research indicates that learning is optimised when it encompasses lower and higher order levels of thinking. Deep learning is enriched by having a knowledge of basic facts and ideas, and vice versa. For example, knowing a range of historical facts enriches deep thinking and learning, which in turn makes information more memorable. Therefore, by giving children a powerful tool for memorising basic information, perhaps their learning outcomes could improve.

I believe that this book highlights the importance of being able to represent concepts and ideas in visual forms. Also, I believe that there are huge benefits from children having a better understanding and experience of how we learn. I have had great fun with my class (and in an assembly) creating memory palaces. I suspect that these techniques could, given consideration about their use, be used to improve learning outcomes for many children.